Robert Daniel Carmichael

From THE LOGIC OF DISCOVERY, 1930 by R. D. Carmichael

     §1   The Nature of Postulates.   It is not so much my purpose to answer this question as to give more general currency among thinkers to the practice of asking it. It seems clear that postulate systems have been employed fully and consciously in only a small part of that domain of thought in which their use will yield characteristic values that perhaps cannot be realized without them. Too much of what is known on the subject is now known to a relatively small group of thinkers, whereas the ideas involved are of such sort as to deserve wide currency and to be useful to thinkers in widely separated fields of thought. Thence arises my desire to give them greater currency. Certain main features of the question have been recently discussed by J. Rueff in an illuminating way in his book, Des Sciences physiques aux Sciences morales, published by Félix Alcan, Paris, in 1922. This book will be of value to those who deal with the so-called exact sciences and of great value to those who are concerned with the ethical and the economic sciences.

Postulates were frequently known as axioms : and axioms were held to be self-evident fundamental truths. In this sense they appeared in Euclid's Elements. The attitude toward them was profoundly affected by the non-euclidean geometries ; there has been a progressive change in the judgment concerning their significance ; and today we see them in a light very different from that which illuminated them for the ancients. We shall not attempt to trace the stages of this development. For our purposes it is sufficient to have in mind a clear conception of what is meant today by a postulate system. Every body of logically connected doctrine must have a starting-point from which the deductions may proceed. This basic material must contain propositions which are taken as assumptions, that is, are not proved. Such propositions are postulates. A fruitful connected consistent body of such propositions constitutes a postulate system. They afford the basis from which deduction may begin. So far as deduction is concerned they are entirely unproved. In the natural sciences there is sometimes direct or indirect empirical evidence for their truth ; and in mathematics there is an esthetic guide to their formation so that in some cases they may be far removed from experience with the external world. In all cases considerations of comfort to eh thinker himself enter into their formation and enunciation.

Not only must we have unproved propositions in a deductive body of logically connected doctrine ; we must also have undefined terms. These terms are substantive or relational. A substantive term is one pointing to an existent thing which may be either material or mental, as for instance the term point in certain postulate systems for geometry. A relational term names a relation holding for two or more terms : as, for example, the relation of betweenness when one point is said to lie between two other points. While such terms are undefined in the sense that no explicit definition is or can be given to them in the system in which they appear they are not unrestricted in meaning. They appear in the fundamental propositions of the postulate system ; hence in every concrete instance of the doctrine, or "doctrinal function", built on these propositions the undefined terms must have such significance as to verify, or render "true", the basic propositions in which they appear. Hence, while they are not and cannot be defined explicitly, they are nevertheless such as to verify the postulate system. Sometimes this may leave a great range of variation open to them ; at other times it may restrict them in a comprehensive way, as in the case of those postulate systems which are called categorical. An interesting discussion of this whole matter in a form accessible to the "educated layman" is to be found in Lectures I to IX of C. J. Keyser's Mathematical Philosophy, published by Dutton, New York, 1922.

Typical examples of postulate systems are to be found in mathematics. Those which occur in geometry are perhaps the ones most readily accessible to the layman. It is in mathematics that the notion of postulate system has been most carefully developed and utilized most consciously and most widely. Fortunately the ideas connected with the whole treatment of postulate systems require for their understanding no more knowledge of mathematics than is usually the possession of a college-trained person. To appreciate them requires a certain intellectual maturity and acumen, but very little technical mathematics. That they are thus accessible to the educated layman is made abundantly clear in Keyser's Mathematical Philosophy. It seems that the way is now open for postulate systems to take their rightful place in the further progress of exact thought in all fields of intellectual activity.

Chicago-London : Open Court 1930, pp. 46-49.

Bibliographic ( University of California http://melvyl.cdlib.org/ )

Author Carmichael, Robert Daniel, 1879- Title The theory of numbers, and Diophantine analysis Publisher New York, Dover Publications [1959] Format Book

Author Carmichael, Robert Daniel, 1879-1967 Title The logic of discovery, by R. D. Carmichael Publisher Chicago, London, The Open Court Pub. Co., 1930 Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967. Title The calculus, by Robert D. Carmichael, James H. Weaver and Lincoln LaPaz. Publisher Boston, Ginn, 1937. Series Textbooks in mathematics Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967. Title What is man? A poem. Publisher Urbana, University of Illinois Press [1950] Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967. Title Introduction to the theory of groups of finite order / by Robert D. Carmichael. Publisher New York, N.Y. : Dover Publications, 1956, c1937. Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967 Title Introduction to the theory of groups of finite order, by Robert D. Carmichael.. Publisher New York, Dover Publications, Inc. [1956, c1937] Format Book Government document

Author Carmichael, R. D. (Robert Daniel), 1879-1967 Title The logic of discovery / by R. D. Carmichael Publisher Chicago : London : The Open Court Pub. Co., 1930 Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967 Title Mathematical tables and formulas, compiled by Robert D. Carmichael and Edwin R. Smith. Publisher New York, Dover Publications [1962, c1931] Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967. Title The logic of discovery / by R. D. Carmichael. Publisher New York : Arno Press, 1975 [c1930] Series History, philosophy and sociology of science Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967. Title Introduction to the theory of groups of finite order, by Robert D. Carmichael. Publisher Boston, New York [etc.] Ginn and Company [c1937] Series Textbooks in mathematics Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967. Title The theory of relativity, by Robert D. Carmichael. Publisher New York, John Wiley & sons, inc.; [etc., etc.] 1913. Series Mathematical monographs, ed. by M. Merriman and R. S. Woodward. no. 12 Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967. Title The theory of relativity, by Robert D. Carmichael ... Publisher New York, John Wiley & Sons, inc.; [etc., etc.] 1920. Series Mathematical monographs, no. 12 Format Book

Author Carmichael, R. D. (Robert Daniel), 1879-1967. Title The theory of numbers, by Robert D. Carmichael. Publisher New York, J. Wiley & sons, inc.; [etc., etc., 1914] Series Mathematical monographs. Ed. by M. Merriman and R. S. Woodward. no.13 Format Book